Most machine learning methods provide only point estimates when being queried to predict on new data. This is problematic when the data is corrupted by noise, e.g. from imperfect measurements, or when the queried data point is very different to the data that the machine learning model has been trained with. Probabilistic modelling in machine learning naturally equips predictions with corresponding uncertainty estimates which allows a practitioner to incorporate information about measurement noise into the modelling process and to know when not to trust the predictions. A well-understood, flexible probabilistic framework is provided by Gaussian processes that are ideal as building blocks of probabilistic models. They lend themself naturally to the problem of regression, i.e., being given a set of inputs and corresponding observations and then predicting likely observations for new unseen inputs, and can also be adapted to many more machine learning tasks. However, exactly inferring the optimal parameters of such a Gaussian process model (in a computationally tractable manner) is only possible for regression tasks in small data regimes. Otherwise, approximate inference methods are needed, the most prominent of which is variational inference.
In this dissertation we study models that are composed of Gaussian processes embedded in other models in order to make those more flexible and/or probabilistic. The first example are deep Gaussian processes which can be thought of as a small network of Gaussian processes and which can be employed for flexible regression. The second model class that we study are Gaussian process state-space models. These can be used for time-series modelling, i.e., the task of being given a stream of data ordered by time and then predicting future observations. For both model classes the state-of-the-art approaches offer a trade-off between expressive models and computational properties (e.g. speed or convergence properties) and mostly employ variational inference. Our goal is to improve inference in both models by first getting a deep understanding of the existing methods and then, based on this, to design better inference methods. We achieve this by either exploring the existing trade-offs or by providing general improvements applicable to multiple methods.
We first provide an extensive background, introducing Gaussian processes and their sparse (approximate and efficient) variants. We continue with a description of the models under consideration in this thesis, deep Gaussian processes and Gaussian process state-space models, including detailed derivations and a theoretical comparison of existing methods.
Then we start analysing deep Gaussian processes more closely: Trading off the properties (good optimisation versus expressivity) of state-of-the-art methods in this field, we propose a new variational inference based approach. We then demonstrate experimentally that our new algorithm leads to better calibrated uncertainty estimates than existing methods.
Next, we turn our attention to Gaussian process state-space models, where we closely analyse the theoretical properties of existing methods. The understanding gained in this process leads us to propose a new inference scheme for general Gaussian process state-space models that incorporates effects on multiple time scales. This method is more efficient than previous approaches for long timeseries and outperforms its comparison partners on data sets in which effects on multiple time scales (fast and slowly varying dynamics) are present.
Finally, we propose a new inference approach for Gaussian process state-space models that trades off the properties of state-of-the-art methods in this field. By combining variational inference with another approximate inference method, the Laplace approximation, we design an efficient algorithm that outperforms its comparison partners since it achieves better calibrated uncertainties.
